Growing random uniform d-ary trees

TL;DR

This paper introduces a new linear-time method for generating uniformly random d-ary trees with n internal nodes, extending R{é}my's binary tree algorithm to arbitrary d-ary trees.

Contribution

It presents a novel construction method for uniform d-ary trees that is inspired by R{é}my's binary algorithm but adapted for general d-ary trees, with linear complexity.

Findings

Provides a linear-time algorithm for uniform d-ary tree generation

Extends binary tree algorithms to d-ary trees

Offers a practical method for random tree sampling

Abstract

Let T d (n) be the set of d-ary rooted trees with n internal nodes. We give a method to construct a sequence (t n , n $\ge$ 0) where, for any n $\ge$ 1, t n has the uniform distribution in T d (n), and t n is constructed from t n--1 by the addition of a new node, and a rearrangement of the structure of t n--1. This method is inspired by R{\'e}my's algorithm which does this job in the binary case, but it is different from it. This provides a method for the random generation of a uniform d-ary tree in T d (n) with a cost linear in n.